Weighted volume monotonicity and isoperimetric comparison under a lower Ricci curvature bound
Abstract
We establish a weighted extension of the Bishop-Gromov volume comparison theorem for complete Riemannian manifolds with Ricci curvature bounded below. Given a point p and positive radial weights f and h, we consider the weighted volume of a geodesic ball and a corresponding model weighted volume in the simply connected space form of constant sectional curvature k. We prove that, under the assumption that h/f is nondecreasing, the ratio of these two weighted volumes is nonincreasing with respect to the radius. We also characterize the equality case, showing that equality forces the weight ratio to be constant and the corresponding geodesic ball to be isometric to the model ball. As applications, we derive weighted Bishop-Gromov-type comparisons, annular comparison inequalities, and weighted volume doubling estimates. We further apply the monotonicity formula to obtain weighted area-volume inequalities and sharp weighted isoperimetric-type comparisons for geodesic balls.
Related
- Eigenvalue comparison theorems for the Witten-Laplacian and the weighted p-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
- Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow
- Monotonicity and evolution of total isoperimetric profiles under Minkowski regularization
- Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity
- The Cartan-Hadamard isoperimetric inequality under controlled variation of curvature
BibTeX
@misc{park2026weighted,
title = {{Weighted volume monotonicity and isoperimetric comparison under a lower Ricci curvature bound}},
author = {Jiewon Park and Keomkyo Seo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17055},
eprint = {2609.17055},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17055},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weighted volume monotonicity and isoperimetric comparison under a lower Ricci curvature bound
AU - Jiewon Park
AU - Keomkyo Seo
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17055
UR - https://arxiv.org/abs/2609.17055
AB - We establish a weighted extension of the Bishop-Gromov volume comparison theorem for complete Riemannian manifolds with Ricci curvature bounded below. Given a point p and positive radial weights f and h, we consider the weighted volume of a geodesic ball and a corresponding model weighted volume in the simply connected space form of constant sectional curvature k. We prove that, under the assumption that h/f is nondecreasing, the ratio of these two weighted volumes is nonincreasing with respect to the radius. We also characterize the equality case, showing that equality forces the weight ratio to be constant and the corresponding geodesic ball to be isometric to the model ball. As applications, we derive weighted Bishop-Gromov-type comparisons, annular comparison inequalities, and weighted volume doubling estimates. We further apply the monotonicity formula to obtain weighted area-volume inequalities and sharp weighted isoperimetric-type comparisons for geodesic balls.
ER -